Problem: The city limits of Las Pythagoras form a perfect shape of an isosceles right triangle whose legs are both $25$ kilometers long. The population in Las Pythagoras is $100{,}000$ people. What is the population density of Las Pythagoras? Round your answer, if necessary, to the nearest integer.
Answer: This is a density word problem. To solve it, we can use the following equation, which is the area definition of density: ${\text{Density}}=\dfrac{{\text{Total quantity}}}{{\text{Area}}}$ What do we know? The city has a shape of an isosceles right triangle with a side length of $25$ kilometer (we can use this to find the ${\text{area}}$ ). The city's population is ${100{,}000}$. This is the ${\text{total quantity}}$. What do we need to find? The city's population ${\text{density}}$. The ${\text{area}}$ is $\dfrac{1}{2}\cdot 25^2={312.5}$ square kilometers. Now we can plug ${\text{total quantity}=100{,}000}$, and ${\text{area}=312.5}$ in the equation. $\begin{aligned} {\text{Density}}&=\dfrac{{\text{Total quantity}}}{{\text{Area}}} \\\\ &=\dfrac{{100{,}000}}{{312.5}} \\\\ &=320 \end{aligned}$ The population density in Las Pythagoras is $320$ people per square kilometer.